Title: Stable maps with p-fields on a smooth projective variety

Abstract: The quantum Lefschetz hyperplane principle is the collective name for various statements relating the quantum cohomology of an ambient space X to that of a smooth subvariety Z, when Z is defined by the vanishing of a regular section of a bundle E. It relates some E-twisted Gromov-Witten theory of X to the usual Gromov-Witten theory of Z. It plays a fundamental role in the proofs of mirror symmetry given by both Givental and Lian-Liu-Yau. In genus 0, with the additional assumption that E is convex, the statement has been reduced to an instance of functoriality of virtual pullbacks and proved by Kim-Kresch-Pantev. However, a naive generalization of this statement does not hold for higher genus. Various replacements have been proposed. Chang-Li introduced a twisted theory on X, the theory of stable maps with p-fields, and showed that it is equal (up to a sign) to the usual Gromov Witten theory of Z. Their original proof concerns the special case of the quintic surface in P^4. I will present my ongoing work to generalize this statement to a smooth projective variety X.